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| Natura: | Recurso digital |
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Zenodo
2026
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| Accesso online: | https://doi.org/10.5281/zenodo.19084943 |
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Sommario:
- <p>Paper 140 validated the quintic compiler at rank 1, verifying the five-factor grammar on the curve 37a1 (ε = −1, gate open). This paper completes the rank 0 channel by running the compiler on 11a1: the elliptic curve y² + y = x³ − x² − 10x − 20, which is the modular curve X₀(11) and the curve of smallest conductor in the Cremona database. The parity gate is closed (ε = +1): the archimedean ψ-mode (ε_∞ = −1) is cancelled by the split multiplicative ψ-mode at 11 (w₁₁ = −1), giving ε = (−1)(−1) = +1. No zero is forced at s = 1, and L(E, 1) ≠ 0 confirms rank 0. The five-factor grammar produces L(E, 1) = Ω · c₁₁ · |Sha| / |E(ℚ)_tors|² = 1.269209 × 5 × 1 / 25 = Ω/5. The rational BSD quotient is 1/5 — the quintic filter's simplest output. The number 5 pervades the curve: conductor 11 = L₅ (the fifth Lucas number, equal to φ⁵ + ψ⁵), torsion order 5, Tamagawa number 5, Kodaira type I₅ (five Néron components), and isogenies of degree 5 and 25 within the isogeny class. The compiler's first curve is tuned entirely to the quintic. With both rank 0 and rank 1 channels validated, the compiler passes every test case where BSD is a theorem.</p>